Transfinite almost square Banach spaces

TL;DR

This paper introduces transfinite almost square Banach spaces, linking their properties to the containment of isomorphic copies of $c_0(\

Contribution

It develops transfinite versions of almost square Banach spaces and connects them to $c_0(\

Findings

Establishes relations between transfinite almost square spaces and $c_0(\

Provides examples and stability results for these properties

Solves an open question related to diameter two properties and octahedral norms

Abstract

It is known that a Banach space contains an isomorphic copy of $c_0$ if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose to relate them to the containment of isomorphic copies of $c_0(\kappa)$, where $\kappa$ is some uncountable cardinal. We also provide several examples and stability results of the above properties by taking direct sums, tensor products and ultraproducts. By connecting the above properties with transfinite analogues of the strong diameter two property and octahedral norms, we obtain a solution to an open question from [8].